From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior

摘要

The study of neuronal interactions is at the center of several big collaborativeneuroscience projects (including the Human Connectome Project,the Blue Brain Project, and the Brainome) that attempt to obtain a detailedmap of the entire brain. Under certain constraints, mathematicaltheory can advance predictions of the expected neural dynamics basedsolely on the statistical properties of the synaptic interaction matrix. Thiswork explores the application of free random variables to the study oflarge synaptic interaction matrices. Besides recovering in a straightforwardway known results on eigenspectra in types of models of neuralnetworks proposed by Rajan and Abbott (2006), we extend them to heavy-tailed distributions of interactions. More important, we analyticallyderive the behavior of eigenvector overlaps, which determine thestability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged,their stability dramatically decreases due to the strong nonorthogonalityof associated eigenvectors. This leads us to the conclusion that understandingthe temporal evolution of asymmetric neural networks requiresconsidering the entangled dynamics of both eigenvectors and eigenvalues,which might bear consequences for learning and memory processesin these models. Considering the success of free random variables theoryin a wide variety of disciplines, we hope that the results presentedhere foster the additional application of these ideas in the area of brainsciences.